People come to help her.

I'm having a hard time understanding the usage of "come (or go )+ verb". For example:

"People are friendly where she lives. When she is sick, people come to help her."

Is "people come to help her" natural English? Would you also say: "people come help her" or "people come and help her"?

I learned in my last post that "go to wash your hands" sounds odd. And "go wash your hands" or "go and wash your hands" is a command.

If I relate this to the above example sentence, the theory would be: "come to help her" is odd; and "come help her" or "come and help her is imperative? But I feel like I'm totally wrong! Or maybe it's all contextual, no hard and fast rule at all?

You are right - like in many (most?) cases, context matters. And in your last post, you were talking about imperatives. This is not an imperative.

'When she is sick, people come to help her' is perfectly natural and correct. You could also say ' . . . people go to help her', but by using 'come' you are sort of writing about it from 'her' perspective.

' . . . people come help her.' would be wrong here too, but ' . . . peoplecome and help her.' is OK.

I see a subtle difference between . . . come to help her' and ' . . . come and help her.'. In the first case, the people come to comfort her or care for her, whereas in the second they come to help her do something.